Question

Let μ be a signed measure on (r,b) with μ(]-[infinity],x])=e2x-ex. Compute the value of ∫ e⁻ˣ² dμ(x) and ∫ e⁻ˣ² d|μ|(x).

Answer 1

To compute the value of the integrals ∫ e⁻ˣ² dμ(x) and ∫ e⁻ˣ² d|μ|(x), we need to substitute the measure into the integrals and simplify the exponent. We can then use integration techniques to compute the integrals.

Step-by-step explanation:

To compute the value of the integral ∫ e⁻ˣ² dμ(x), we need to substitute the measure μ(]-[infinity],x])=e2x-ex into the integral. This gives us ∫ e⁻ˣ² e2x-ex dx. Simplifying the exponent, we get ∫ eˣ(2x-x²) dx. To compute the integral, we can use integration techniques such as integration by parts or substitution.

To compute the value of the integral ∫ e⁻ˣ² d|μ|(x), we need to substitute the absolute value of the measure |μ|(]-[infinity],x])=e2x-ex into the integral. This gives us ∫ e⁻ˣ² e2x-ex dx. Again, simplifying the exponent, we get ∫ eˣ(2x-x²) dx. We can use the same integration techniques to compute this integral as well.